Python scripting: how to select next vertex in mesh
You can get all adjacent vertices like this 

import bpy
import bmesh

obj = bpy.context.active_object
mesh = obj.data

bm = bmesh.from_edit_mesh(mesh)
active = bm.select_history.active

if active and isinstance(active, bmesh.types.BMVert):
    for edge in active.link_edges:
        print(edge.other_vert(active))